Algebraic set definitions and subsets (Higher)

Two signs in the same bus station window. One says free travel for under sixteens. The other says free travel for under eighteens. The inspector wants to know whether he has to check both, or just one. Just one. Everybody under sixteen is automatically under eighteen, so that whole group already sits inside the other. Now change the second sign to students, and it falls apart: a twenty year old student is not under eighteen. Same shaped question, opposite answer. Telling those two apart is this whole topic.

One flag before we start. This is Higher tier content. If you are sitting Foundation, you can safely skip this one, and the video on Venn diagrams, the universal set, the empty set and the complement is the one you want instead. If you are on Higher, stay. Two ideas here, and they fit together neatly.

First idea: how these sets get written down. Not as a list of things. As a set of instructions.

Straight off a paper. A equals, open curly bracket, x, colon, x is a multiple of four, comma, x is less than or equal to twenty, close curly bracket. The colon is read as such that. The comma is read as and. Out loud: A is the set of all values of x, such that x is a multiple of four and x is less than or equal to twenty. That x is not something to solve for. It is a slot. Any number can be dropped in, and the two conditions decide whether it stays. A rule earns its place. It stays exact when the list would be huge, and it hands you a test you can run on any number at all. Passes both conditions, in. Fails either one, out.

So run it. The numbers under discussion are the whole numbers from one to twenty, so those are the only candidates. One is not a multiple of four, out. Two, out. Three, out. Four is a multiple of four, and four is less than twenty. Both conditions pass, so four is in. After that it is every fourth number. Eight, in. Twelve, in. Sixteen, in. Then twenty, and twenty is the one worth slowing down for. The condition says less than or equal to twenty. Less than, or equal to. Twenty is equal to twenty, so twenty passes that test, and twenty is a multiple of four. It goes in. Had the paper printed a plain less than sign, twenty would be out. One short line underneath, different answer. Zero stays out too, and not because it fails the multiple of four test. Zero passes that one. Zero is out because it was never a candidate. The numbers under discussion start at one. So A is four, eight, twelve, sixteen and twenty. Curly brackets, in order, each element written once.

Your go, same shape. B equals x such that x is even and x is less than or equal to twenty, from those same candidates, one to twenty. Which listing of B is right. Option one: zero, two, four, and so on up to twenty. Option two: two, four, six, up to eighteen. Option three: two, four, six, up to twenty. Pick one. I'll wait. Option three. It runs all the way to twenty, because twenty is even and twenty is equal to twenty. Option two stopped at eighteen, which is the answer to a plain less than sign. Option one opened with zero, which was never a candidate. Read the symbol, then check where the candidates start.

Second idea. A symbol that compares two whole sets in one go.

It is written A, then a rounded shape opening towards B, with a short line underneath, then B. You say it as: A is a subset of B. And it claims something strict. Every element of A is also in B. Not most of them. Every one. Two details in that symbol earn their keep. The open side faces the set doing the containing, so the symbol shows you which way the claim points. And the line underneath behaves like the line under a less than or equal to sign. It allows A to be the whole of B.

Back to A, and here is B. B equals x such that x is even and x is less than or equal to twenty. So B is two, four, six, eight, ten, twelve, fourteen, sixteen, eighteen and twenty. Is A a subset of B. Give a reason. Slow way first. Four is in B. Eight is in B. Twelve, sixteen and twenty, all in B. Five elements of A, five hits. So yes, A is a subset of B. That is the answer. It is not yet the reason. A list of hits only covers the five numbers you happened to check. Here is the reason. A multiple of four is four times something. Four is two times two, so four times something is two times something else, and that is exactly what even means. Every multiple of four is even. Not just these five. All of them. Draw that and you do not get two overlapping circles. You get one circle sitting entirely inside another. A, the multiples of four, drawn wholly inside B, the even numbers. Nothing pokes out. That picture is what subset means.

This is the question the whole topic runs on, asked a new way. Which ones do I keep. Element by element, you have been answering it one at a time. A subset asks it about a whole set at once, and if the answer for every element of A is keep it, then A was already inside B.

Now the part that catches people out. The direction the claim points.

Second pair. C is x such that x is a factor of twelve. D is x such that x is a factor of twenty four. List both, then test both directions. Factors come in pairs, so hunt them in pairs. One times twelve, two times six, three times four. C is one, two, three, four, six and twelve. Now twenty four. One times twenty four, two times twelve, three times eight, four times six. D is one, two, three, four, six, eight, twelve and twenty four. Is C a subset of D. Run down C. One, two, three, four, six, twelve, and every one of them is in D. Yes. And the reason is not the tick list. Twenty four is twelve times two, so anything that divides twelve divides twenty four as well. Now turn it round. Is D a subset of C. Is every factor of twenty four also a factor of twelve. Have a look down D first. No. Eight is a factor of twenty four, and eight is not a factor of twelve. That one number is enough. A subset claim says every element, so a single failure destroys it. Twenty four itself does the same job. C sits inside D. D does not sit inside C.

Say that plainly, because it is the trap here. Subset is a one way claim. The two sets look related, so it feels like it ought to work both ways. It does not. Two directions, two separate checks, every time.

And when both directions do hold, nothing has gone wrong. It means the two sets contain exactly the same elements. They are the same set, written two different ways.

Now how these arrive on the paper. Take a question like this: two instructions in a row. List the elements of A. Then: is A a subset of B, give a reason. Two instructions, two separate jobs. A correct list with no reason written underneath leaves the second job not done. And a reason is a sentence, not a repeat of the list. Not: yes, I checked them all. Something like: yes, because every multiple of four is also even. It has to cover every element, not only the ones you happened to write down. Two things to check before you commit. First, the inequality symbol. Less than, or less than or equal to. Say it out loud, and look at where the numbers under discussion start and stop, because that decides which numbers are candidates at all. Second, which way the subset symbol faces. Read it as a full sentence, every element of this set is in that set, and test that sentence. One failing element is enough to disprove it, and write that element down.

Your turn, both halves at once. The numbers under discussion are the whole numbers from one to thirty. P is x such that x is a multiple of six and x is less than or equal to thirty. Q is x such that x is a multiple of three and x is less than or equal to thirty. List P, then test both directions, with a reason for each. Take your time. I'll wait right here. P is six, twelve, eighteen, twenty four and thirty. Thirty is in, because thirty is equal to thirty. Is P a subset of Q. Yes, and here is why: six is three times two, so every multiple of six is a multiple of three. Is Q a subset of P. No. Three is a multiple of three and is not a multiple of six. One counter example, and the claim is gone.

Time to fix this in the shape you will use it. What you see printed, and what it should make you do. You see curly brackets with a colon inside. That is a rule, not a list. Build the list by testing candidates one at a time, out of the numbers under discussion, and say the inequality symbol out loud before you decide the last one. You see the subset symbol. That is a claim that every element of the left set is in the right one. To agree with it, give a reason that covers every element. To disagree, name one element that fails. And it points one way only, so the reverse is a separate check. Which ones do I keep. For a subset, all of them.

Next in the chapter: three-set Venn diagrams, element counts and the n of A notation, on the Higher tier. Three circles instead of two, and notation for how many elements a set holds.

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